On some classes of almost periodic functions in abstract spaces

PII. S016117120440653X http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON SOME CLASSES OF ALMOST PERIODIC FUNCTIONS

IN ABSTRACT SPACES

DARIUSZ BUGAJEWSKI and GASTON M. N’GUÉRÉKATA

Received 15 August 2004

We deal withC(n)_{-almost periodic functions taking values in a Banach space. We give} sev-eral properties of such functions, in particular, we investigate their behavior in view of differentiation as well as integration. The superposition operator acting in the space of such functions is also under consideration. Some applications to ordinary as well as par-tial differenpar-tial equations are presented. Moreover, we introduce the class of the so-called asymptoticallyC(n)-almost periodic functions and give some of their properties.

2000 Mathematics Subject Classification: 42A75, 34C27.

1. Introduction. C(n)_{-almost periodic functions are one of the most important} eralizations of the concept of almost periodic functions in the sense of Bohr. This gen-eralization relies on the requirement that a given function as well as its derivatives up to thenth order inclusively are almost periodic in the sense of Bohr. Many properties of such functions with real values are given in [1,2].

In the present paper, we deal withC(n)_{-almost periodic functions taking values in} a Banach space. This work is organized in the following way: inSection 2, we give the definition of vector-valuedC(n)_{-almost periodic functions and we prove that the} lin-ear space of such functions considered with a suitable norm turns out to be a Banach space. Moreover, we give an application to an ordinary linear homogeneous differential equation, where the corresponding operator A on the right side is linear and com-pact. InSection 3, we investigate the behavior ofC(n)_{-almost periodic functions with} respect to differentiation as well as integration. Further, we prove that theC(n)_-almost periodicity of the initial functions f and g imply that the solutions of the classical wave equations in view of the first variable areC(n)_{-almost periodic. Next, inSection 4,} we investigate the superposition operator acting in the space ofC(n)_{-almost periodic} functions. In particular, we give examples of such functions and we consider the case of functions of two variables. Further, inSection 5, we introduce a new class of the so-called asymptoticallyC(n)_{-almost periodic functions and give several of their} prop-erties.

2. Definitions and basic properties. Let(E,·)be a Banach space. We will use the following notation:fh(x)=f (h+x), wheref:R→Eandh, x∈R.

(briefly:C(n)_{(E)) consisting of such functionsf:R→Efor which}

sup t∈R

f (t)+

n

i=1

f(i)_(t)

<+∞, (2.1)

wheref(i)_{denotes theith derivative off. In the spaceC}(n)

B (R, E)(briefly:C (n) B (E)) we introduce the norm

fn=sup t∈R

f (t)+

n

i=1

f(i)_(t)

forf∈CB(n)(E). (2.2)

It is well known thatCB(n)(E)with this norm is a Banach space.

Definition2.1. A numberτ ∈Ris said to be a ( · n, ε)-almost period (briefly: ( · n, ε)-a.p.) of a function f ∈C(n)_{(E), if f−fτ}

n ≤ε for ε >0. The set of all (·n, ε)-almost periods of a functionf will be denoted byE(n)_{(ε;f ).}

Definition_2.2. _{A functionf}∈_C(n)(R, E)_{is said to beC}(n)_{-almost periodic (briefly:} C(n)_{-a.p.), if for everyε >0 the setE}(n)_{(ε;f )is relatively dense inR.}

It can be proved easily using the continuity off(i)_{, i=1,2, . . . , n, that this set of} (·n, ε)-almost periods is a closed set.

Denote byAP(n)_{(E)the set of allC}(n)_{-a.p. functions.}

Directly from the above definitions it follows thatAP(n+1)_(E)⊂AP(n)_{(E). Moreover,} puttingn=0, we haveAP(0)_{(E)=AP (E), whereAP (E)denotes the classical Banach} space of all almost periodic functionsR→E, in the sense of Bohr.

Now, we prove the following lemma.

Lemma2.3. AP(n)(E)⊂C_B(n)(E).

Proof. Letf∈AP(n)(E)and letε >0. Fort∈R, we have

f (t)+

n

i=1

f(i)_{(t)≤f (t)−fτ(t)+} n

i=1

f(i)_(t)−f(i)

τ (t)+fτ(t)+ n

i=1

f(i) τ (t).

(2.3)

Fixt∈R. Sincef∈AP(n)_{(E), there exists(·n, ε)-a.p.τ=(ε, t)∈(−t,−t+l), where} l=l(ε) >0 is a number which characterizes the relative density ofE(n)_{(ε;f ). Moreover,} sincef∈C(n)_{(E), there existsM >0 such that}

max s∈[0,l]

f (s)+

n

i=1

f(i)(s)

≤M, (2.4)

In what follows, we will need the following well-known lemma.

Lemma2.4. Letf , g∈AP (E). Then, for everyε >0there exists a relatively dense set

of their commonε-a.p.

An immediate consequence ofLemma 2.4is the following criterion.

Theorem2.5. f∈AP(n)(E)if and only iff(i)∈AP (E)fori=0,1, . . . , n.

By the above result, we infer that iff∈AP(n)_{(E), then the range off}(i)_{is relatively} compact for eachi=0,1, . . . , n(see [7, Remark 3.1.6, page 53]).

Now, we recall the very important Bochner’s criterion.

Theorem_2.6. _{A continuous functionf:}R→_{E is almost periodic if and only if for} every sequence of real numbers(sn), there exists a subsequence(sn)such that(f (t+sn))

converges uniformly int∈R.

See, for example, [7, Theorem 3.1.8, page 55].

It is clear that based on Theorem 2.6, we are able to formulate an analogue of Bochner’s criterion for the case ofC(n)_{-almost periodicity.}

Theorem2.7. A functionf∈C(n)(E)isC(n)-almost periodic if and only if for every

sequence of real numbers(sn), there exists a subsequence(sn)such that(f(i)(t+sn))

converges uniformly int∈Rfori=0,1, . . . , n.

Corollary_2.8. _{Let(T (t))t∈Rbe aC0-group of bounded linear operators onEand}

for ane∈E, the functionxe:R→Edefined byxe(t)=T (t)esatisfies the assumptions:

(1) xe∈C(n)_{(E)for somen;}

(2) Rxeis relatively compact inE, whereRxe denotes the range of the functionxe.

Then,xe∈AP(n)_(E).

Proof. It is an immediate consequence ofTheorem 2.7 and the group property.

Application2.9. Consider in a Banach spaceEthe abstract differential equation

x(t)=Ax(t) fort∈R, (2.5)

whereA∈L(E)is a compact linear operator. Letx(t)be a solution of the above equa-tion.

Then, the following hold:

(1) ifRxis bounded, thenx∈APn_{(E)for alln=1,2, . . .,}

(2) ifRxis relatively compact, thenx∈APn_{(E)for alln=0,1,2, . . . .}

Proof. Clearly,x(t)=etAx(0)∈Cn(E), for alln=0,1,2, . . ., sinceAis continuous and defined on allE.

To prove assertion (2), it suffices to observe that x(t)=etA_{x(0); thus obviously} x∈APn_{(E)for alln=0,1,2, . . ., byCorollary 2.8.}

We have also the following simple proposition.

Proposition_2.10. _{A linear combination ofC}(n)_{-a.p. functions is aC}(n)_{-a.p. function.}

Moreover, letEbe a Banach space over the fieldK (K=RorC). Iff ∈AP(n)_(E)and

ν∈AP(n)_{(K), then the following functions are also inAP}(n)_(E): (i) νf,

(ii) fa(t):=f (t+a), wherea∈Ris fixed, (iii) h(t):=f (−t),t∈R.

Proof. It is a simple consequence ofTheorem 2.7.

Theorem2.11. If a sequence(fk), k∈N, ofC(n)-a.p. functions is such that fk−

fn→0ask→ ∞, thenfisC(n)-a.p. function.

Proof. _{Fixε >0. There existsk0}∈N_{such thatf}−_fk

0n≤ε/3. Forτ∈E

(n)_{ε/3;

fk0}we obtain

f−fτ_n≤f−fk_0n+fk₀−fk_0τn+fk_0τ−fτ_n≤ε, (2.6)

soE(n)_(ε/3;fk

0)⊂E(n)(ε;f ), and thusf∈AP(n)(E).

Corollary_2.12. _AP(n)_{(E)considered with the norm (2.2) turns out to be a Banach}

space.

Proof. In view ofProposition 2.10and Lemma 2.3, AP(n)(E)is a linear subspace ofCB(n)(E). Let(fk), fk∈AP(n)(E),k∈N, be a Cauchy sequence. Then, there exists f∈CB(n)(E)such that limk→∞fk−fn=0. ByTheorem 2.11,f∈AP(n)(E), so it is a Banach space.

3. Differentiation and integration. The first result in this Section gives a sufficient condition which guarantees that the derivative of a function f ∈AP(n)_{(E)is also a} C(n)_{-a.p. function.}

Theorem3.1. Iff∈AP(n)(E)andf(n+1)is uniformly continuous, thenfisC(n)-a.p.

function.

Proof. _{Fixε >0. In view ofTheorem 2.5and the uniform continuity off}(n+1)_{, there}

existsδ >0 such that forh∈Rwith|h|< δwe have

n+1

i=1

f(i)_(t)−f(i)

Further, fort∈Randh≠0, we have

f (t+h)−f (t)

h −f

_(t)=1

h

h

0

f(s+t)−f(t)ds,

fh_h−f−f n =sup t∈R 1 h h 0

f(s+t)−f(t)ds+

n

i=1

di dti 1 h h 0

f(s+t)−f(t)ds

≤sup t∈R 1 h h 0  n+1

i=1

d

i

dtif (s+t)−f (i)_(t)

 ds.

(3.2)

Thus, by (3.1), we obtain(fh−f )/h−fn≤εfor eachh∈Rwith|h|< δ. Now, we applyTheorem 2.11and the proof is complete.

We recall now the well-known Bohl-Bohr-type theorem (see, e.g., [7, Theorem 3.2.6, page 62]).

Theorem_{3.2(see [7, Theorem 3.2.6, page 62])}. _Letf∈_{AP (E)and let the function}

F:R→Edefined byF (t)=0tf (s)ds; thenF∈AP (E)if and only if its rangeRF= {F (t): t∈R}is relatively compact inE.

One can extend this result in the following way.

Theorem 3.3. If f ∈ AP (E) and the range RF is relatively compact, then F ∈

AP(1)_(E).

Proof. In view of Bohl-Bohr theorem,F∈AP (E). Moreover, sincef is continuous, F(t)=f (t)fort∈R. Thus,F∈AP (E)and, consequently,F∈AP(1)_(E).

Theorem 3.3is a special case of the following theorem.

Theorem 3.4. If f ∈AP(n)(E) and the rangeRF is relatively compact, then F ∈

AP(n+1)_(E).

Proof. In view of Bohl-Bohr theorem,F∈AP (E). Moreover,F=f ∈AP(n)(E). In view ofTheorem 2.5, we getF∈AP(n+1)_(E).

Corollary_3.5. _Iff∈_AP(n)_{(E)and the rangeRf is relatively compact, thenf}∈

AP(n+1)_(E).

Proof. Sincefis continuous onR, so,

f (t)=f (0)+

t

0f

_{(s)ds fort∈R. (3.3)}

In view ofTheorem 3.4,f∈AP(n+1)_(E).

Example3.6. Letf∈AP(n)(E), and for a fixeda∈R, define a functionF:R→Eby Fa(t):=t+a

Then,F∈AP(n)_{(E). Moreover,fandF have the same(·n, ε)-almost periods.} First, we note that by a suitable change of variable we getFa(t)=a

0f (t+ξ)dξ. It is then easy to check thatF∈C(n)_(E).

Now, letτ∈E(n)_{(/|a|, f ); then, for anyt∈R, we have}

_{Fa(t+τ)−Fa(t)n≤}a 0

_{f (t+τ+ξ)−f (t+ξ)ndξ≤ |a|}

|a|=, (3.4)

that isτ∈E(n)_{(/|a|, F ), which proves our assertion.}

Example_3.7. _{Consider the wave equation}

uxx(x, t)=utt(x, t), u(x,0)=f (x), ut(x,0)=g(x). (3.5)

Assume that bothf , g∈AP(n)_{(R). Then, for eacht0∈R, the solutionu(x, t0)∈AP}(n)_. Indeed, we have

ux, t0=1₂fx+t0+fx−t0+1₂

x+t0

x−t0

g(s)ds. (3.6)

Clearly,(1/2)(f (x+t0)+f (x−t0))∈AP(n)_{(R)byProposition 2.10. Also, if we write}

x+t0

x−t0g(s)ds=

0

x−t0g(s)ds+

x+t0

0 g(s)ds, we deduce that it is inAP(n)(R)by the pre-vious example.

4. Superposition operator. LetE1,E2be Banach spaces. We begin this section with the following proposition.

Proposition_4.1. _IfA:E1→_{E2is bounded linear operator andf}∈_AP(n)_{(E1), then} Af∈AP(n)_(E2).

Proof. It is clear that(Af )(n)=Af(n). Then, it is enough to applyTheorem 2.5and [7, Corollary 2.1.6, page 14] for almost periodic functions.

For everyf∈AP(n)_{(E1), define theE2-valued functionGas follows:}

G(t):=

t

0

Af (s)ds, (4.1)

whereA:E1→E2is a bounded linear operator. We have the following corollary.

Corollary_4.2. _LetA:E1→_{E2be a bounded linear operator with relatively compact}

range, then theE2-valued functionGdefined above is inAP(n+1)_(E2).

Proof. According toProposition 4.1,Af∈AP(n)(E2)for everyf∈AP(n)(E1). We also note that RG ⊂R(A), where R(A) is the range of the operator A, hence RG is relatively compact. We now complete the proof usingTheorem 3.4.

Now, we will consider the superposition operator (the autonomous case) acting on the spaceAP(n)_(E1).

We will utilize the fact that the rangeRf = {f (t)|t∈R}of aC(n)_{-a.p. function is} relatively compact to prove the following theorem.

Theorem4.3. Ifφ∈C(n)(E1, E2)andf∈AP(n)(E1), thenφ◦f∈AP(n)(E2).

Proof. _{First, we observe that the result holds for n}=_{0 (see [7, Theorem 2.1.5,} page 14] for almost periodic functions). Now, we show that it holds for n=1. We have(φ◦f )(x)=φ(f (x))◦f(x)forx∈E1. Sincef∈AP (E1)=AP (L(R, E1))and φ◦f∈AP (L(E1, E2)), so(φ◦f )∈AP (E2)=AP (L(R, E2)). Suppose the result holds forn−1. This implies that φ◦f ∈AP(n−1)_{(L(E1, E2)). Moreover, f∈AP}(n−1)_(E1), so (φ◦f )∈AP(n−1)_{(E2). Moreover,Rφ}

◦f is relatively compact, so, byTheorem 3.4, φ◦f∈AP(n)_(E2).

Example _4.4. _{Having Theorem 4.3, one can easily produce examples ofC}(n)_-a.p. functions. For example, leth(x)=exp{f (x)}, wheref (x)=sin(αx)+sin(βx),x∈R andα,βare incommensurate real numbers.

Example_4.5. _{It is also easy to give examples ofC}(n)_{-a.p. functions which are not}

C(n+1)_{-a.p. ones. For example, let}

f (x)=

        

x−2_πk

2

sin 1

x−2k/π ifx∈

_2k−2

π , 2k+1

π

\

_2k

π

,

0 ifx=2_πk,

(4.2)

wherek=0,±1,±2, . . . .One can easily verify thatfis periodic with a periodTf =2/π andf is continuous onR. Thus,f∈AP (E). Now, we observe thatf(x)exists at each x∈Randfis not continuous atx=2k/π, wherek=0,±1,±2, . . . (cf. [5]). Thus,f does not belong toAP(1)_(R).

The functionfcan be utilized to construct an example for arbitraryn≥2 (see [1]).

Now, we will deal with functions of two variablesf (t, x). First, recall the following definition from [4] in a Banach space setting.

Definition4.6. A continuous functionf:R×E→E is said to be almost periodic intfor eachx∈Eif for eachε >0 there existsl >0 such that every interval[a, a+l] contains at least one pointssuch that

f (t+s, x)−f (t, x)< ε, for eacht∈Rand for eachx∈E. (4.3)

In view of Bochner’s criterion (Theorem 2.6), it is clear that the above definition is equivalent to the following one.

Based onTheorem 2.5, we can introduce the following definition.

Definition4.8. A functionf:R×E→E, (t, x)→f (t, x), is said to be (n)-almost periodic intfor eachx∈Eif(∂i_{f /∂t}i_{)(t, x)is almost periodic fori=0,1, . . . , n} (ap-parently∂0_{f /∂t}0_=f).

An analogue ofTheorem 2.7is the following one.

Theorem_4.9. _Letf:R×_E→_{Ebe such that(∂}i_{f /∂t}i_{)(t, x)are continuous fori}=

0,1, . . . , n. Then,fis (n)-almost periodic if and only if for every sequence of real numbers (sn), there exists a subsequence(sn)such that(D

(i)

1 f (t+sn, x))converges uniformly in t∈R andx∈E,i=0,1, . . . , n, whereD(i)1 f denotes theith partial derivative off in view of the first variable.

Remark_4.10. _{We consider the nonautonomous composition operator acting in the} spaceAP(n)_{(E). First, we recall the corresponding result forAP (E)(see [4, Lemma 3.8]} in Fréchet space setting).

Letf:R×E→Ebe almost periodic intfor eachx∈Rand assume thatfsatisfies a Lipschitz condition inxuniformly int∈R, that is,f (t, x)−f (t, y) ≤Lx−yfor allt∈Randx, y∈E. Letϕ:R→Ebe almost periodic. Then, the functionF:R→E defined byF (ϕ)(t)=f (t, ϕ(t))is almost periodic.

We notice that the assumptions of the above result imply thatF satisfies a (global) Lipschitz condition as a superposition operator acting in the spaceC(R, E).

Recall that in the case of the space C(n)_{(R, E)the superposition operator F} satis-fies the global Lipschitz condition F (x1)−F (x2)n≤kx1−x2n (k >0, x1, x2∈ C(n)_{(R, E)) if and only if the functionf has the formf (s, u)=a(s)+b(s)ufor some} a, b∈C(n)_{(R, E)(see [3, Theorem 8.4, page 212] or [6]). If we assume additionally that} f:R×E→Eis (n)-almost periodic intfor eachx∈E, then it is equivalent to assume thata, b∈AP(n)_{(R). In this situation,F acts inAP}(n)_{(E), obviously.}

5. AsymptoticallyC(n)_{-almost periodic functions}

Definition5.1. A functionf∈C(n)(R₊, E)is said to be asymptoticallyC(n)-almost periodic if it admits a decomposition

f (t)=g(t)+h(t), t∈R, (5.1)

whereg:R→EisC(n)_{-almost periodic function andh∈C}(n)_(R

+, E)with limt→+∞h(t) =0. Thus,gand hare called, respectively, the principal and corrective terms of the functionf.

As an obvious consequence of the above definition andProposition 2.10, we have the following proposition.

Proposition _5.2. _{Letf1, f2, f :}R₊ →_{E, ν:}R₊→_{R be asymptoticallyC}(n)_-almost

periodic. Then,f1+f2,λf for anyλ∈R andνf are also asymptoticallyC(n)_-almost

Now, we prove the following important theorem.

Theorem5.3. The decomposition of aC(n)-a.p. function is unique.

Proof. The idea of the proof is similar to that from the proof of [7, Theorem 2.5.4]. Letf:R+→EbeC(n)-almost periodic with two decompositions:

f (t)=gi(t)+hi(t), t∈R+, i=1,2, (5.2)

with principal termsg1,g2and corrective termsh1,h2, respectively. Then, fort∈R+ we have

g1(t)−g2(t)+h1(t)−h2(t)=0, (5.3)

so in view ofDefinition 5.1,

lim t→+∞

g1(t)−g2(t)=0. (5.4)

Consider the sequence(n). Sinceg1−g2isC(n)_{-almost periodic, in view ofTheorem 2.7} we can extract a subsequence(nk)of(n)such that

lim k→∞

g1t+nk−g2t+nk=F (t) (5.5)

uniformly int∈R. Thus, by (5.4),F (t)=0 fort∈R. Since

0=lim k→∞F

t−nk=g1(t)−g2(t), (5.6)

h1(t)=h2(t)=0 fort∈R. The proof is complete.

In view ofProposition 5.2, we can consider the set of allC(n)_{-a.p. functionsR}

+→Eas a linear space. We will denote it byAAP(n)_{(E). Moreover, byTheorem 5.3, the formula}

fAn= gn+hn, (5.7)

wheregand hare the principal and corrective terms of the functionf, respectively, defines the norm on the spaceAAP(n)_{(E)(obviously in the case of the norm of the} corrective termh, the supremum is taken overR+). We have the following theorem.

Theorem5.4. AAP(n)(E)considered with the norm (5.7) is a Banach space.

respect to the norm of the spaceAP(n)_{(E). Thus, byCorollary 2.8, there exists aC}(n) -almost periodic functiongsuch that(fn)g−gn→0, asn→ ∞. Further, the corrective terms of the functions fn :(fn)h form a Cauchy sequence in the spaceC(n)_(R

+, E). Hence, there exists a continuous functionh∈C(n)_(R

+, E)such that(fn)h−hn→0, as n→ ∞. Since for everyn∈Nlimt→+∞(fn)h(t)=0 andh(t)=h(t)−(fn)h(t)+(fn)h(t) fort∈R+, limt→+∞h(t)=0. Thus,f:=g+h∈AAP(n)(E)and limn→∞fn−fAn=0, soAAP(n)_{(E)is a Banach space.}

Theorem_5.5. _{LetE1,E2be Banach spaces and letf:}R₊→_{E1be an asymptotically}

C(n)_{-almost periodic function. Letφ:E1→E2be a mapping of the classC}(n)_{. Moreover,}

assume that there exists a compact setB which contains the closure of the set{h(t):

t∈R+}, where his the corrective term of f. Then, the function φ(f (t)):R+→E2 is

C(n)_-a.p.

Proof. Letf (t)=g(t)+h(t)fort∈R₊, wheregandhare principal and corrective terms off, respectively. In view ofTheorem 4.3, the functionφ(g(t))isC(n)_-almost periodic. SetΓ(t)=φ(f (t))−φ(g(t))fort∈R+. Obviously,Γ is aC(n)-mapping. Let

ε >0. By assumption, there exists a compact set C which contains the closures of {f (t):t∈R+}and{g(t):t∈R+}, soφrestricted toCis uniformly continuous. Thus, there existsδ=δ(ε) >0 such that

φ(x)−φ(y)₂< ε ifx−y1< δ, x, y∈E1, (5.8)

where·1,·2denote the norms ofE1andE2, respectively. Since limt→+∞h(t)=0, there existst0>0 such that

f (t)−g(t)₁=h(t)₁< δ fort > t0. (5.9)

Thus, we obtain

_Γ(t)2=φ

f (t)−φg(t)2< ε fort > t0. (5.10)

Hence, the functionφ(f (t))=φ(g(t))+Γ(t)is asymptoticallyC(n)_-a.p.

The last result of this section concerns the integration of asymptoticallyC(n)_-almost periodic functions.

Theorem5.6. LetEbe a Banach space andf:R₊→Ean asymptoticallyC(n)-almost

periodic function. Consider the function F :R+ →E defined by F (t)=0tf (s)ds and

G: R→E defined by G(t)= t

0g(s)ds, where g is the principal term off. Assume

G has a relatively compact range in E and that ₀+∞h(t)dt <+∞, whereh is the

corrective term off. Then,F is asymptoticallyC(n)_{-almost periodic; its principal term is}

G(t)+0+∞h(s)dsand its corrective term isH(t)= −

+∞

t h(s)ds.

and limt→+∞H(t)=0. Since

F (t)=G(t)+

_+∞

0 h(s)ds+H(t) fort∈R+. (5.11)

The proof is complete.

Acknowledgment. We would like to thank Dr. Toka Diagana for all his comments.

References

[1] M. Adamczak,C(n)-almost periodic functions, Comment. Math. Prace Mat.37(1997), 1–12. [2] M. Adamczak and S. Stoínski,On the(NC(n)_{)-almost periodic functions, Proceedings of the}

6th Conference Function Spaces (R. Grzáslewicz, Cz. Ryll-Nardzewski, H. Hudzik, and J. Musielak, eds.), World Scientific Publishing, New Jersey, 2003, pp. 39–48.

[3] J. Appell and P. P. Zabrejko,Nonlinear Superposition Operators, Cambridge Tracts in Math-ematics, vol. 95, Cambridge University Press, Cambridge, 1990.

[4] D. Bugajewski and G. M. N’Guérékata,Almost periodicity in Fréchet spaces, J. Math. Anal. Appl.299(2004), no. 2, 534–549.

[5] H. and J. Musielakowie,Mathematical Analysis, Vol. I, WN UAM, Pozna´n, 1993.

[6] J. Matkowski,Form of Lipschitz operators of substitution in Banach spaces of differentiable functions, Zeszyty Nauk. Politech. Łódz. Mat. (1984), no. 17, 5–10.

[7] G. M. N’Guérékata,Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, 2001.

Dariusz Bugajewski: Faculty of Mathematics and Computer Science, Adam Mickiewicz Univer-sity, Umultowska 87, Pozna´n 61-614, Poland

E-mail address:[email protected]

Gaston M. N’Guérékata: Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA